Groupe d’études et de recherche en analyse des décisions

Construction of lattice rules for multiple integration based on a weighted discrepancy

Vasile Sinescu

In this talk, I am looking to give a brief overview on the construction of lattice rules that have what is termed a "low weighted discrepancy". In simple terms, the "discrepancy" is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called "product weights". In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are "general". In the weighted settings mentioned above, the quality of the lattice rules is assessed by the "weighted discrepancy" mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges from \(O(n^{-1/2})\) to the (believed) optimal \(O(n^{-1+\delta})\) for any \(\delta>0\), with the involved constant independent of the dimension.